To understand virtual manipulatives, first a description of a manipulative (also known as a physical manipulative) is necessary. A manipulative is any device that helps a student's understanding of an abstract mathematical idea. Students often struggle with abstract ideas. The manipulative allows a student take a hands-on approach to learning. Manipulatives provide students and teachers a way to represent concretely the abstract concepts that they are learning and activate prior knowledge. Examples are tangrams, interlocking blocks, Dienes blocks, counters, geoboards, fraction kits, base ten blocks, pattern blocks, geo strips, and fraction strips. Manipulatives are believed to have been around since ancient times, dating back to a counting device called the abacus used by the Babylonians in 300 B.C. (Carruccio, 2006). With advances in technology, manipulatives began to be reproduced through Web-based, computer representations called virtual manipulatives. Virtual manipulatives are defined as “interactive, web-based representations of a dynamic object that presents opportunities for constructing mathematical knowledge” (Moyer, Bolyard & Spikell, 2002, p. 373). These manipulatives are still concrete, though they are not “physical” (Clements and McMillen, 1996). Of course, students cannot physically "pick up" a virtual manipulative but they can interact with a virtual manipulative by "picking it up" on the computer screen. Students can construct meaning on their own by using the mouse to control physical actions of objects by sliding, flipping, turning, and rotating them. Teachers see the potential for virtual manipulatives as a huge aide to their work. They can be of great use in the classroom because they are “more manageable, clean, flexible, and extensible” (Clements and McMillen, 1996, p. 271). Their interactive capabilities make virtual manipulatives a valuable tool. Because students are acting on these manipulatives, they are able to derive meaning and relationships for mathematical concepts (Moyer, Bolyard & Spikell, 2002). As students become more and more adept with new technologies, they may find virtual manipulatives more fun and exciting and become more motivated to learn.
Important Findings on Student Outcomes
Research shows that virtual manipulatives help students develop more complex understandings than they can develop with physical manipulatives. Students who used virtual manipulatives, especially English Language Learners (ELL), found it easier to explain their learning (Moyer, Niezgoda, and Stanley, 2005). ELL students also can benefit by being able to choose English, Spanish, or French as the language for the virtual manipulative display.
Clements and McMillen (1996) found that less freedom to move virtual manipulatives vs. physical manipulatives may be more helpful to students. The study explored whether it is too restrictive or too hard for students to operate the on-screen symbols of virtual manipulatives rather than simply operating the physical manipulatives. Findings suggest that students who worked with computer-generated base-ten blocks used symbols more meaningfully, and were able to connect the symbols they wrote on paper to the virtual manipulative blocks. Conversely, students who used physical base-ten blocks had trouble writing symbols to represent the physical blocks. Those students tended to look at the symbols on paper and the physical blocks as two different activities (p. 274).
In a study of a middle school learning unit of fractions, virtual manipulatives were found to be more time-efficient than physical manipulatives (Mendiboro and Hasselbring 2011). Also, students reported that they liked using virtual manipulatives due to the immediate feedback they received. For instance, using a virtual manipulative that allows students to manipulate sizes of triangles, students could immediately see that the angles of a triangle always added up to 180 degrees.
Emerging Trends and Open Issues
As with most classroom learning in this age of emerging technology, teachers are still the key to learning when virtual manipulatives are implemented. Although physical manipulatives and virtual manipulatives are found to be valuable for helping students learn, manipulatives by themselves have no inherent meaning. Therefore, it is up to the teachers to guide students into building connections between the manipulatives and the abstract symbols they represent.
In research findings, Mendiboro and Hasselbring (2011) point out that despite three decades of recommendations by the National Council of Teachers of Mathematics to use manipulatives in the classroom, teachers are still reluctant to use physical manipulatives because of practical and pedigogically difficulties. But because virtual manipulatives so closely mirror instruction using physical manipulatives, there is a strong chance that teachers will find that virtual manipulatives overcome the difficulties associated with physical manipulatives and use them more in the future.
Virtual manipulatives have an advantage over physical manipulatives in that they give students on-screen hints and feedback, which physical manipulatives cannot without the help of teachers. Although virtual manipulatives can aid students through individual use, students still need teachers' help to use them correctly and make connections between the representations and the math (Suh and Moyer, 2007). Another advantage is that teachers can also involve the whole class in an interactive virtual manipulative lesson by using them with a Smart board or interactive whiteboards (Mildenhall, Swan, Northcote, and Marshall, 2008). Still, teachers need to become experts at using virtual manipulatives for them to be an effective learning tool. Those teachers who aren't adept at using them are "more likely to use them as a diversionary rather than an instructional activity" (Moyer, 2001).
Bouck, Emily C. & Flanagan, Sara M. (2010). Virtual Manipulatives: What they are and how teachers can use them. Intervention in School and Clinic, 45 (3), 186-191.
Carruccio, Ettore (2006), Mathematics And Logic in History And in Contemporary Thought. Piscataway, N.J.: Aldine Transaction.
Clements, D. H., & McMillen, S. (1996). Rethinking “concrete” manipulatives. Teaching Children Mathematics, 2, 270-279.
Hodge, T., & Brumbaugh, D. (2003). Web-based manipulatives.Teaching Children Mathematics, 9 (8), 461.
Mendiburo, Maria, & Hasselbring, Ted (2011). Technology’s Impact on Fraction Learning: An experimental comparison of virtual and physical manipulatives. SREE Conference Abstract Template. Mildenhall, P., Swan, P., Northcote, M. & Marshall, L. (2008). Virtual manipulatives on the interactive white board: A preliminary investigation. Australian Primary Mathematics Classroom, 13(1), 9-14.
Moyer, P. S. (2001). Are we having fun yet? How teachers use manipulatives to teach mathematics. Educational Studies in Mathematics, 47(2), 175-197.
Moyer, P.S., Bolyard, J. J., & Spikell, M. A. (2002). What are virtual manipulatives? Teaching Children Mathematics, 8(6), 372-377.
Moyer, P. S., Niezgoda, D., & Stanley, J. (2005). Young children’s use of virtual manipulatives and other forms of mathematical representations. In W. J. Masalski & P.C. Elliott (Eds.), Technology-supported mathematics learning environments: Sixty-seventh yearbook (pp. 17-34). Reston, VA: National Council of Teachers of Mathematics.
Pitcher, Heather (2007). Math Education: Manipulatives. HeatherPitcher.BlogSpot.com. Web resource. Suh, J., & Moyer, P. S. (2007). Developing students’ representational fluency using virtual and physical algebra balances. Journal of Computers in Mathematics and Science Teaching, 26(2), 155-173. Reviewed By: (Paul Richards, Megan Parcell)
Virtual Manipulatives
David TippenhauerDescription and Uses of Technology
To understand virtual manipulatives, first a description of a manipulative (also known as a physical manipulative) is necessary. A manipulative is any device that helps a student's understanding of an abstract mathematical idea. Students often struggle with abstract ideas. The manipulative allows a student take a hands-on approach to learning. Manipulatives provide students and teachers a way to represent concretely the abstract concepts that they are learning and activate prior knowledge. Examples are tangrams, interlocking blocks, Dienes blocks, counters, geoboards, fraction kits, base ten blocks, pattern blocks, geo strips, and fraction strips. Manipulatives are believed to have been around since ancient times, dating back to a counting device called the abacus used by the Babylonians in 300 B.C. (Carruccio, 2006).
With advances in technology, manipulatives began to be reproduced through Web-based, computer representations called virtual manipulatives. Virtual manipulatives are defined as “interactive, web-based representations of a dynamic object that presents opportunities for constructing mathematical knowledge” (Moyer, Bolyard & Spikell, 2002, p. 373). These manipulatives are still concrete, though they are not “physical” (Clements and McMillen, 1996). Of course, students cannot physically "pick up" a virtual manipulative but they can interact with a virtual manipulative by "picking it up" on the computer screen. Students can construct meaning on their own by using the mouse to control physical actions of objects by sliding, flipping, turning, and rotating them.
Teachers see the potential for virtual manipulatives as a huge aide to their work. They can be of great use in the classroom because they are “more manageable, clean, flexible, and extensible” (Clements and McMillen, 1996, p. 271). Their interactive capabilities make virtual manipulatives a valuable tool. Because students are acting on these manipulatives, they are able to derive meaning and relationships for mathematical concepts (Moyer, Bolyard & Spikell, 2002). As students become more and more adept with new technologies, they may find virtual manipulatives more fun and exciting and become more motivated to learn.
Important Findings on Student Outcomes
Research shows that virtual manipulatives help students develop more complex understandings than they can develop with physical manipulatives. Students who used virtual manipulatives, especially English Language Learners (ELL), found it easier to explain their learning (Moyer, Niezgoda, and Stanley, 2005). ELL students also can benefit by being able to choose English, Spanish, or French as the language for the virtual manipulative display.
Clements and McMillen (1996) found that less freedom to move virtual manipulatives vs. physical manipulatives may be more helpful to students. The study explored whether it is too restrictive or too hard for students to operate the on-screen symbols of virtual manipulatives rather than simply operating the physical manipulatives. Findings suggest that students who worked with computer-generated base-ten blocks used symbols more meaningfully, and were able to connect the symbols they wrote on paper to the virtual manipulative blocks. Conversely, students who used physical base-ten blocks had trouble writing symbols to represent the physical blocks. Those students tended to look at the symbols on paper and the physical blocks as two different activities (p. 274).
In a study of a middle school learning unit of fractions, virtual manipulatives were found to be more time-efficient than physical manipulatives (Mendiboro and Hasselbring 2011). Also, students reported that they liked using virtual manipulatives due to the immediate feedback they received. For instance, using a virtual manipulative that allows students to manipulate sizes of triangles, students could immediately see that the angles of a triangle always added up to 180 degrees.
Emerging Trends and Open Issues
As with most classroom learning in this age of emerging technology, teachers are still the key to learning when virtual manipulatives are implemented. Although physical manipulatives and virtual manipulatives are found to be valuable for helping students learn, manipulatives by themselves have no inherent meaning. Therefore, it is up to the teachers to guide students into building connections between the manipulatives and the abstract symbols they represent.
In research findings, Mendiboro and Hasselbring (2011) point out that despite three decades of recommendations by the National Council of Teachers of Mathematics to use manipulatives in the classroom, teachers are still reluctant to use physical manipulatives because of practical and pedigogically difficulties. But because virtual manipulatives so closely mirror instruction using physical manipulatives, there is a strong chance that teachers will find that virtual manipulatives overcome the difficulties associated with physical manipulatives and use them more in the future.
Virtual manipulatives have an advantage over physical manipulatives in that they give students on-screen hints and feedback, which physical manipulatives cannot without the help of teachers. Although virtual manipulatives can aid students through individual use, students still need teachers' help to use them correctly and make connections between the representations and the math (Suh and Moyer, 2007). Another advantage is that teachers can also involve the whole class in an interactive virtual manipulative lesson by using them with a Smart board or interactive whiteboards (Mildenhall, Swan, Northcote, and Marshall, 2008). Still, teachers need to become experts at using virtual manipulatives for them to be an effective learning tool. Those teachers who aren't adept at using them are "more likely to use them as a diversionary rather than an instructional activity" (Moyer, 2001).
The Center for Implementing Technology in Education lists several resources for finding virtual manipulatives on the Internet:Arcytech - Educational Java Programs; Illuminations; National Library of Virtual Manipulatives for Interactive Mathematics;Project Interactivate; and MathTools.
References
Bouck, Emily C. & Flanagan, Sara M. (2010). Virtual Manipulatives: What they are and how teachers can use them. Intervention in School and Clinic, 45 (3), 186-191.
Carruccio, Ettore (2006), Mathematics And Logic in History And in Contemporary Thought. Piscataway, N.J.: Aldine Transaction.
Clements, D. H., & McMillen, S. (1996). Rethinking “concrete” manipulatives. Teaching Children Mathematics, 2, 270-279.
Hodge, T., & Brumbaugh, D. (2003). Web-based manipulatives.Teaching Children Mathematics, 9 (8), 461.
Mendiburo, Maria, & Hasselbring, Ted (2011). Technology’s Impact on Fraction Learning: An experimental comparison of virtual
and physical manipulatives. SREE Conference Abstract Template.
Mildenhall, P., Swan, P., Northcote, M. & Marshall, L. (2008). Virtual manipulatives on the interactive white board: A preliminary investigation. Australian Primary Mathematics Classroom, 13(1), 9-14.
Moyer, P. S. (2001). Are we having fun yet? How teachers use manipulatives to teach mathematics. Educational Studies in Mathematics, 47(2), 175-197.
Moyer, P.S., Bolyard, J. J., & Spikell, M. A. (2002). What are virtual manipulatives? Teaching Children Mathematics, 8(6), 372-377.
Moyer, P. S., Niezgoda, D., & Stanley, J. (2005). Young children’s use of virtual manipulatives and other forms of mathematical representations. In W. J. Masalski & P.C. Elliott (Eds.), Technology-supported mathematics learning environments: Sixty-seventh yearbook (pp. 17-34). Reston, VA: National Council of Teachers of Mathematics.
Pitcher, Heather (2007). Math Education: Manipulatives. HeatherPitcher.BlogSpot.com. Web resource.
Suh, J., & Moyer, P. S. (2007). Developing students’ representational fluency using virtual and physical algebra balances. Journal of Computers in Mathematics and Science Teaching, 26(2), 155-173.
Reviewed By: (Paul Richards, Megan Parcell)